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14
An InteriorPoint Method for Semidefinite Programming
, 2005
"... We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other appli ..."
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Cited by 207 (17 self)
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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other applications include maxmin eigenvalue problems and relaxations for the stable set problem.
Method of centers for minimizing generalized eigenvalues
 Linear Algebra Appl
, 1993
"... We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fr ..."
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Cited by 65 (14 self)
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We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fractional programs. Many problems arising in control theory can be cast in this form. The problem is nondifferentiable but quasiconvex, so methods such as Kelley's cuttingplane algorithm or the ellipsoid algorithm of Shor, Nemirovksy, and Yudin are guaranteed to minimize it. In this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently. The algorithm is a variation on Huard's method of centers, using a selfconcordant barrier for matrix inequalities developed by Nesterov and Nemirovsky. (Nesterov and Nemirovsky have also extended their potential reduction methods to handle the same problem [NN91b].) Since the problem is quasiconvex but not convex, devising a nonheuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper, in some cases, than a more general result due to Nesterov and Nemirovsky. The algorithm is demonstrated on an example: determining the quadratic Lyapunov function that optimizes a decay rate estimate for a differential inclusion.
Semidefinite programs and combinatorial optimization (Lecture notes)
, 1995
"... this paper, we are only concerned about the last question, which can be answered using semidefinite programming. For a survey of other aspects of such geometric representations, see [64]. ..."
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Cited by 30 (1 self)
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this paper, we are only concerned about the last question, which can be answered using semidefinite programming. For a survey of other aspects of such geometric representations, see [64].
ConeLP's and Semidefinite Programs: Geometry and a Simplextype Method
, 1996
"... . We consider optimization problems expressed as a linear program with a cone constraint. ConeLP's subsume ordinary linear programs, and semidefinite programs. We study the notions of basic solutions, nondegeneracy, and feasible directions, and propose a generalization of the simplex method for a l ..."
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Cited by 19 (2 self)
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. We consider optimization problems expressed as a linear program with a cone constraint. ConeLP's subsume ordinary linear programs, and semidefinite programs. We study the notions of basic solutions, nondegeneracy, and feasible directions, and propose a generalization of the simplex method for a large class including LP's and SDP's. One key feature of our approach is considering feasible directions as a sum of two directions. In LP, these correspond to variables leaving and entering the basis, respectively. The resulting algorithm for SDP inherits several important properties of the LPsimplex method. In particular, the linesearch can be done in the current face of the cone, similarly to LP, where the linesearch must determine only the variable leaving the basis. 1 Introduction Consider the optimization problem Min cx s:t: x 2 K Ax = b (P ) where K is a closed cone in R k , A 2 R m\Thetak ; b 2 R m ; c 2 R k : (P ) is called a linear program over a cone, or a coneLP. It m...
SEMIDEFINITE PROGRAMMING RELAXATIONS OF NONCONVEX PROBLEMS IN CONTROL AND COMBINATORIAL OPTIMIZATION
"... We point out some connections between applications of semidefinite programming in control and in combinatorial optimization. In both fields semidefinite programs arise as convex relaxations of NPhard quadratic optimization problems. We also show that these relaxations are readily extended to optimi ..."
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Cited by 17 (2 self)
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We point out some connections between applications of semidefinite programming in control and in combinatorial optimization. In both fields semidefinite programs arise as convex relaxations of NPhard quadratic optimization problems. We also show that these relaxations are readily extended to optimization problems over bilinear matrix inequalities.
Semidefinite Programming for Assignment and Partitioning Problems
, 1996
"... Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP hard combinatorial optimization problems of simpl ..."
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Cited by 13 (2 self)
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Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP hard combinatorial optimization problems of simple structure such as the maxcut and graph bisection problems. In this work, we try to solve more complicated combinatorial problems such as the quadratic assignment, general graph partitioning and set partitioning problems. A tight SDP relaxation can be obtained by exploiting the geometrical structure of the convex hull of the feasible points of the original combinatorial problem. The analysis of the structure enables us to find the socalled "minimal face" and "gangster operator" of the SDP. This plays a significant role in simplifying the problem and enables us to derive a unified SDP relaxation for the three different problems. We develop an efficient "partial infeasible" primaldual inter...
MaxMin Eigenvalue Problems, PrimalDual Interior Point Algorithms, and Trust Region Subproblems
 Optimization Methods and Software
, 1993
"... Two Primaldual interior point algorithms are presented for the problem of maximizing the smallest eigenvalue of a symmetric matrix over diagonal perturbations. These algorithms prove to be simple, robust, and efficient. Both algorithms are based on transforming the Technische Universitat Graz, I ..."
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Cited by 10 (4 self)
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Two Primaldual interior point algorithms are presented for the problem of maximizing the smallest eigenvalue of a symmetric matrix over diagonal perturbations. These algorithms prove to be simple, robust, and efficient. Both algorithms are based on transforming the Technische Universitat Graz, Institut fur Mathematik, Kopernikusgasse 24, A8010 Graz, Austria. Research support by Christian Doppler Laboratorium fur Diskrete Optimierung. y Program in Statistics & Operations Research, Princeton University, Princeton, NJ 08544. Research support by AFOSR through grant AFOSR910359. z Research support by the National Science and Engineering Research Council Canada. problem to one over the cone of positive semidefinite matrices. One of the algorithms does this transformation through an intermediate transformation to a trust region subproblem. This allows the removal of a dense row. Key words: Maxmin eigenvalue problems, trust region subproblems, Loewner partial order, primaldual ...
Explicit solutions for interval semidefinite linear programs
 Linear Algebra Appl
, 1996
"... We consider the special class of semidefinite linear programs (IV P) maximize traceCX subject to L ≼ A(X) ≼ U, where C, X, L, U are symmetric matrices, A is a (onto) linear operator, and ≼ denotes the Löwner (positive semidefinite) partial order. We present explicit representations for the general ..."
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Cited by 9 (2 self)
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We consider the special class of semidefinite linear programs (IV P) maximize traceCX subject to L ≼ A(X) ≼ U, where C, X, L, U are symmetric matrices, A is a (onto) linear operator, and ≼ denotes the Löwner (positive semidefinite) partial order. We present explicit representations for the general primal and dual optimal solutions. This extends the results for standard linear programming that appeared in BenIsrael and Charnes, 1968. This work is further motivated by the explicit solutions for a different class of semidefinite problems presented recently in Yang and Vanderbei, 1993.